2 Answers
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The dictionary you link to says that parent is usually in the plural, and the example sentences it gives use the plural. It is a regular plural, formed by adding an s. Many times dictionaries will not list the plural form if it is not irregular.

Multiple parents is a term used to acknowledge the different roles and relationships a person can have with a child.

There are several types of parents, however. A person can have, and only have, two biological parents, from which the genetic makeup of the child in derived. [NB: at least at the time of this writing]

There are also step parents who have a legal but not biological relationship with the child.

There are also psychological parents who can have a pastoral or emotionally supportive relationship with the child.

Currently the legal profession is struggling to keep up with the available technology.

One could say pairs of parents, however there is an underlying assumption in that phrase that parents occur in pairs. If a gay or lesbian couple have a child, presumably there would be three parents involved, two of the same sex and one of the opposite sex.